Download Problems with kinematic mean field electrodynamics at high

Mon. Not. R. Astron. Soc. 395, L48–L51 (2009)
doi:10.1111/j.1745-3933.2009.00639.x
Problems with kinematic mean field electrodynamics at high magnetic
Reynolds numbers
F. Cattaneo1 and D. W. Hughes2
1 Department
2 Department
of Astronomy and Astrophysics and the Computation Institute, University of Chicago, Chicago, IL 60637, USA
of Applied Mathematics, University of Leeds, Leeds LS2 9JT
Accepted 2009 February 9. Received 2009 February 6; in original form 2008 May 14
ABSTRACT
We discuss the applicability of the kinematic α-effect formalism at high magnetic Reynolds
numbers. In this regime, the underlying flow is likely to be a small-scale dynamo, leading to
the exponential growth of fluctuations. Difficulties arise with both the actual calculation of the
α coefficients and their interpretation. We argue that although the former may be circumvented
– and we outline several procedures by which the α coefficients can be computed in principle
– the interpretation of these quantities in terms of the evolution of the large-scale field may be
fundamentally flawed.
Key words: magnetic fields – MHD – turbulence.
1 I N T RO D U C T I O N
Mean field electrodynamics was formalized over forty years ago in
a remarkable paper by Steenbeck, Krause & Rädler (1966). Within
its framework, it is possible to derive an equation for the evolution
of a magnetic field on a scale large compared with that of the
velocity. This equation is much simpler to solve than the induction
equation from which it is derived. Consequently, it has had an
enormous influence on dynamo theory to this day.
Strictly speaking, mean field electrodynamics is a kinematic theory that addresses the growth of a weak seed field. Its function is
thus to predict the growth rate and structure of the generated magnetic field. In its simplest form, in which the underlying velocity
is isotropic and homogeneous, the evolution equation for the largescale field depends on two quantities; α, the mean induction, and β,
the turbulent diffusivity. The growth rate of a field of length-scale
1/k is then
s = αk − βk 2
(1)
(Moffatt 1978). The beauty of this result is that, provided α is
non-zero, dynamo growth is guaranteed on sufficiently large scales.
In the kinematic limit, the coefficients α and β are determined
solely by the properties of the velocity and the magnetic Reynolds
number Rm.
Mean field electrodynamics relies on a scale separation between
fluctuating and mean quantities. Provided such a scale separation
can be enforced, the formalism is valid for any value of Rm. One
of the challenges of the theory is to calculate α and β in terms of
properties of the velocity. If Rm is small, this can be rigorously accomplished by the use of what is known as the first-order smoothing
approximation (FOSA); if Rm is large, the determination of the coefficients α and β becomes more problematic. Indeed, there has been
E-mail: [email protected] (FC);
[email protected] (DWH)
considerable discussion over whether results obtained under FOSA
can be extended to the high Rm regime, where the approximation
is not valid (see Moffatt 1978; Krause & Rädler 1980). However,
we believe that there is a more fundamental issue associated with
the high Rm regime, namely the possibility of small-scale dynamo
action. This being the case, we should ask if is it still meaningful to
apply the ideas and formalism of mean field theory to cases in which
the small-scale fluctuations are growing exponentially, irrespective
of the value of α.
In this Letter, we argue that two problems arise. The first is that
the calculation of the coefficients α and β from numerical or experimental data, say, becomes extremely difficult. The presence of
exponentially growing fluctuations introduces very onerous requirements on the size of the samples that are necessary for the accurate
determination of the coefficients α and β. If such requirements are
ignored, the coefficients remain strongly fluctuating functions of
time (e.g. Sur, Brandenburg & Subramanian 2008) and it becomes
conceptually unsatisfactory to pick a value at some particular instant and argue that it is the physically relevant one. The second
more fundamental problem is that the requirement of scale separation that underpins the validity of the mean field approach cannot be
enforced (Boldyrev, Cattaneo & Rosner 2005). Consequently, the
coefficients α and β – even if they can be computed – fail to provide
information about the rate of growth of the large-scale field, which
is precisely the reason they were introduced in the first place. More
specifically, the large-scale averages of the magnetic field grow at
the same exponential rate as the small-scale fluctuations, and not
at the rate predicted by mean field theory through equation (1) or
similar.
In Section 2, we give a brief outline of the formulation of mean
field electrodynamics and the derivation of the α and β coefficients.
In Section 3, we introduce the various methods that are used to
determine α and β. In Section 4, we discuss convergence and issues
relating to the influence of initial conditions and the requirements
on sample size in order to achieve a given accuracy. In Section 5,
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2009 The Authors. Journal compilation Problems with mean field electrodynamics
we discuss the problems that arise with the physical interpretation
of the α and β coefficients at high Rm.
2 F O R M U L AT I O N O F M E A N F I E L D
E L E C T RO DY N A M I C S
The motivation for mean field electrodynamics is to understand the
evolution of a magnetic field on scales large compared with those
of the turbulent velocity responsible for its generation. The starting
point is the magnetic induction equation:
∂B
(2)
= ∇ × (u × B) + η∇ 2 B,
∂t
where B is the magnetic field and u is the velocity. The idea is
to introduce an average over spatial scales intermediate between
the large integral scales and the small scales characteristic of the
velocity, so that
B = B + b;
(3)
we shall assume that the velocity has no large scale. Taking the
average of equation (2) leads to the induction equations for the
mean and fluctuating components of B, namely
∂B
= ∇ × E + η∇ 2 B,
∂t
(4)
∂b
= ∇ × (u × B) + ∇ × G + η∇ 2 b,
(5)
∂t
where E = u × b is the mean electromotive force (emf) and
G = u × b − u ×b. In order to make progress with equation (4),
one needs to relate E to B. In the kinematic limit, in which u is
independent of B, this can be achieved by noting that the linearity
of equation (5) leads to a linear expression of the form (Moffatt
1978)
E i = αij Bj + βij k
∂
Bj + . . . ,
∂xk
(6)
where αij and β ij k are pseudo-tensors dependent on the properties
of the velocity u and on η.
The simplest case to consider, which still captures the essence of
the problem, is that in which the velocity is homogeneous and
isotropic and hence αij and β ij k take the form αij = αδij and
β ij k = β ij k . Here, α is a pseudo-scalar and hence is non-zero only
for turbulence lacking reflectional symmetry; β, on the other hand,
is a true scalar and hence can be non-zero even for reflectionally
symmetric turbulence. Substituting α and β into equation (4) gives
∂B
(7)
= α∇ × B + (η + β)∇ 2 B.
∂t
The interpretation of α and β is straightforward: α represents mean
induction and β is a turbulent diffusivity. The simplest solution of
this equation can be obtained for the case when ∇ × B = kB,
in which case the growth rate is given by equation (1).
It is easy to show that if the magnetic Reynolds number Rm is
small then the G term in equation (5) can be neglected.1 In this case,
the fluctuations in the magnetic field arise solely from interactions
between the velocity and the mean magnetic field; thus for the case
when the mean field B is uniform (and hence constant in time),
these are bounded by some power of Rm. Solutions of equation (5)
1
It can also be neglected if the correlation time of the velocity is sufficiently
small; here, though we shall mainly be concerned with the case when it is
comparable with the turnover time.
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can be readily obtained, giving explicit expressions for α and β.
The practice of neglecting the G term is often referred to as the
quasi-linear approximation or the FOSA.
When Rm is large, one is no longer justified in neglecting the G
term. In this regime, there are no closed form solutions for b and
one typically has to rely on numerical solutions of equation (5).
Once the solutions for b are determined then the coefficients α and
β can be reconstructed. The problem becomes particularly acute
when small-scale dynamo action occurs and the fluctuations grow
exponentially. Later, we shall argue that the onset of small-scale
dynamo action does not just introduce technical difficulties into
the determination of the coefficients α and β, but does, in fact,
undermine the assumption of scale separation and hence the very
foundation of the mean field approach.
3 T E C H N I Q U E S F O R C A L C U L AT I N G α A N D β
In order to fix ideas, let us consider a turbulent isotropic homogeneous flow, which may or may not be helical, at high Rm in a periodic
domain. We assume further that the velocity has a well-defined correlation length and, where necessary, we consider domains of size
much greater than . For such a system, it is important to ask what
can be measured in terms of α and β and, crucially, what can be
inferred about the initial growth of a magnetic field on large scales.
In this section, we shall discuss in detail various approaches that
have been used in the determination of α and β. The first, and most
natural, is to consider an experiment in which a uniform magnetic
field B 0 is applied, in the z-direction, say. In this case,
u × b · ẑ
,
(8)
B0
where denotes a volume average. The necessary size of the
volume will be discussed presently. We note that since u has zero
average then b can be replaced by B in equation (8). The attractive
feature of this procedure is that since B 0 is uniform then scale
separation is guaranteed. Clearly though this method can be used
only to determine α but not β.
In order also to determine β this approach can be modified by
considering large-scale fields with non-zero gradient. From the
definition
α=
E = u × b = αB − β∇ × B,
(9)
it is clear that α and β can be determined through the consideration
of two independent large-scale magnetic fields, yielding two independent measures of E. Here, b and B are defined according to
expression (3) and B is obtained from solution of the full induction
equation (2). A variant of this method, sometimes referred to as the
test field procedure, is obtained by taking B to be a given test
field, not necessarily equal to any actual average of B, and b to be
the solution of the fluctuating equation (5) driven by that specific
test field (Schrinner et al. 2005).
An altogether different procedure can be constructed by considering a Lagrangian approach. If X(a, t) represents the position at
time t of a fluid element initially located at position a then
Ẋ i (a, t) = ui (X(a, t), t),
(10)
where u is the (Eulerian) fluid velocity. The evolution of an infinitesimal line element along the trajectory is given by
δxi (a, t) = Jij (a, t)δxj (a, 0),
(11)
where Jij is the Jacobian satisfying
J̇ ij =
∂ui
(X(a, t)) Jkj ,
∂xk
Jij (a, 0) = δij .
(12)
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F. Cattaneo and D. W. Hughes
The correspondence between expression (11) and the Cauchy solution for a magnetic field in a perfect conductor (i.e. infinite Rm)
led Moffatt (1974) to derive expressions for α and β in terms of
Lagrangian quantities, formally valid when Rm is infinite. The expression for α is given by
to assume that
|b|
|u × b|
∼ |u|
,
σα ∼
|B|
|B|
(15)
where |u × b| and |u| refer to typical values of these quantities.
Here, we are making the reasonable assumption – well verified in
practice by numerical experiments (see e.g. Hughes & Cattaneo
2008) – that at high Rm the average value of the emf is much
smaller than the typical value of the contributions to the average
(the fluctuations), so that the standard deviation can be used as a
reasonable measure of the magnitude of the fluctuations themselves.
If there is no small-scale dynamo action, then
1
(13)
ij k ui (X, t)Jj k (t).
3
The corresponding expression for β is somewhat involved and can
be found as equation (7.116) in Moffatt (1978). The average here
is over trajectories, which can be realized in two ways. One is
for fixed a and many realizations of the velocity; the other is for
many different values of a for a single realization of the velocity.
The former can be regarded as an ensemble average, whereas the
latter can be regarded as a volume average. This procedure can be
extended to the case of finite Rm by adding a randomly fluctuating
delta-correlated velocity to equation (10) (Drummond & Horgan
1986).
α=
|b|
∼ Rmγ1 ,
|B|
(16)
where 1/2 < γ 1 < 1, its value depending on the velocity. This
can be readily converted into an estimate for the linear size of the
domain needed for accurate computation of the averages, namely
|u|Rmγ1 2/3
.
(17)
L∼
4 INFLUENCE OF INITIAL CONDITIONS
AND SAMPLE SIZE
The situation becomes more complicated when small-scale dynamo
action occurs and |u × b| increases exponentially at the small-scale
dynamo growth rate. For the cases where the large-scale field is
either uniform or held constant (the test field case), the contributions
to the average grow exponentially with the small-scale dynamo
growth rate. Here, the requisite linear size of the domain takes the
form
|u|Rmγ2 est 2/3
(18)
L∼
All of the procedures above require evolving the fluctuating magnetic field to a certain time and then taking either a volume average
or an average over trajectories. It is important that the average is
taken after sufficient time has elapsed such that there is no influence of the initial conditions. This will typically take a few turnover
times, the precise value depending on the initial condition for b,
on the velocity u and on Rm. Two cases must be distinguished. In
one case, Rm is below the threshold for small-scale dynamo action;
here, the volume integrals involved in the measurement of the emf
must be taken only after b2 , say, has become stationary. In the
other case, the small-scale dynamo is operative and eventually, after some time ts , will cause the fluctuations to grow exponentially
with a well-defined growth rate s. The value of ts is determined
by the initial conditions, and related to the time it takes to generate
fluctuations at the diffusive scale. Spatial averages taken before time
ts has elapsed will vary with time and depend on the specific initial
conditions. Thus, meaningful measurements can be taken only after
b2 exp (−2st) has become stationary (cf. Sur et al. 2008).
The above considerations apply equally to either the Eulerian
or Lagrangian methods provided Rm is finite. In the Lagrangian
approach with no stochastic component (Rm infinite), there are
no eigenfunctions and the different moments of b will grow at
different rates. A possible strategy is to integrate until the flux
shows a well-defined exponential growth. Interestingly, according
to the flux conjecture (Finn & Ott 1990), this growth rate is identical
to that of the fast dynamo growth rate in the limit as Rm → ∞.
The size of the volume over which averages are taken is also
crucially important, since ultimately it determines the size of the
error. In order to estimate the required volume, we make use of the
relation for the standard error of the mean (see e.g. Rice 1995):
2
σ
;
(14)
N∼
for some O(1) exponent γ 2 .
In the case when B is itself time-dependent, the estimate for L
depends on the magnitude of the ratio of the fluctuating emf to that
of the mean field, both of which are growing exponentially. With
this method, both the fluctuating field b and the mean field B are
obtained from the solution of the induction equation (2); there is a
single dynamo growth rate and therefore the fluctuating and mean
fields will eventually grow at exactly the same rate. In this case, L
is similar to that in equation (17), but with a different O(1) exponent, γ 3 .
The above discussion has been couched in terms of volume averages. For the cases when the contributions to the averages are
bounded, one can conceive of combining volume and time averages. However, when the contributions are growing exponentially,
such a procedure is not tenable.
For the Lagrangian method, the equivalent issue concerns the
number of independent trajectories needed to determine α. The
source of concern here is that in expression (13) the Jacobian,
on average, grows exponentially, with a rate given by the largest
Lyapunov exponent. So, once again, the number of independent
trajectories needed in the averaging procedure grows exponentially.
5 W H AT D O E S I T A L L M E A N ?
The considerations above suggest that the coefficients α and β can,
at least in principle, always be computed. However, in practice, the
size of the computational domain (or the number of trajectories)
required may be extremely large, and indeed, in some cases when
small-scale dynamo action is present, may even be increasing exponentially in time. Nevertheless, if the procedures are carried out
correctly, i.e. so that convergence can be demonstrated, they should
all give the same answers.
here, is the (desired) uncertainty in the mean of N independent
samples each with sample standard deviation σ – thus given and
σ , N follows. If the turbulent velocity has a characteristic eddy size
, then it is reasonable to assume that the emf has a comparable
characteristic scale; N then denotes the number of patches of size that are needed to achieve the required accuracy. To proceed further,
we need an estimate for σ . For the evaluation of α, it is reasonable
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We now turn to the question of how α and β relate to the growth
of a dynamo field. First consider the case when no small-scale
dynamo action is observed. Here, α/β (i.e. k) must be small; for,
if it were not, then, by equation (1), one would observe dynamo
action on a scale – contrary to our assumption. One can conceive
of a sequence of experiments of varying spatial extent L. When L ∼
, no dynamo action is observed, by assumption. As L is increased,
dynamo action will first set in when L = β/α. Increasing L further
will lead to an increase in the growth rate until its maximum is
reached at L = 2β/α. Further increases in L will not lead to any
further increases in the growth rate. So in the absence of small-scale
dynamo action everything is fine, and α and β determine the growth
rate of the observed magnetic structures.
Consider now the case when small-scale dynamo action is possible. What would be the outcome of repeating the experiments
described above? By assumption, dynamo action takes place even
when L ∼ . Furthermore, the dynamo growth rate will be independent of domain size. Finally, any average of the magnetic field on
intermediate scales will grow at exactly the same rate. Crucially,
the growth rate of the observed field has nothing to do with that
predicted by equation (1). A particularly striking example can be
seen for the case of a non-helical dynamo. Here, α is zero and
equation (1) therefore predicts that large-scale averages should decay exponentially at a scale-dependent rate, whereas, as we have
just argued, any average will grow exponentially at the small-scale
dynamo growth rate.
As argued above, the validity of the mean field approach breaks
down when small-scale dynamo action occurs, which one anticipates at high Rm. Interestingly, as pointed out by Moffatt (1978),
there are problems in the high Rm limit even within the mean field
formalism, since requiring that α/β be small is inconsistent with
traditional estimates for the turbulent α-effect (α ∼ u) and the turbulent β-effect (β ∼ u). In this case, mean field theory predicts
by equation (1) that the fastest growing ‘mean field’ has the same
scale as the fluctuations – namely a small-scale dynamo. It should
be noted, however, that the growth rate of this small-scale dynamo
predicted by mean field theory is not the correct one since it relies
on lack of reflectional symmetry whereas the actual growth rate
does not.
All the considerations above address the kinematic evolution of
magnetic fields, which formally is the relevant regime for mean
field electrodynamics. However, there have been attempts to extend the mean field approach to the non-linear regime. We conclude this Letter by considering a particular case in which many
of the issues we have discussed are pertinent. It is possible to consider a case in which the velocity, rather than being prescribed,
is the self-consistent solution of a saturated small-scale dynamo.
Here, both the velocity and magnetic fluctuations are stationary,
and it is therefore possible, at least in principle, to measure α
and β. In practice, this process is tricky since any finite amplitude perturbation, such as the introduction of a mean field, will
induce a corresponding change in the velocity. Putting aside the
not inconsiderable technical difficulties involved in calculating α
and β, it behoves us to ask whether these quantities convey useful
information about magnetic field evolution. Clearly, they do not
tell us very much about the fluctuations, since in the kinematic
regime the fluctuations grow at the small-scale dynamo growth
rate which is determined by neither α nor β, and in the dynamical
regime the fluctuations are assumed to be stationary anyway. One
could, however, hope that α and β could tell us something about
the evolution of a large-scale perturbation superposed on a sea of
fluctuations.
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To make these ideas concrete, we consider a very specific case.
First suppose that a numerical experiment is conducted on a periodic
domain of size L with L ∼ ; we assume that small-scale dynamo
action is observed, the magnetic field grows to a finite amplitude
and saturates, and, by whatever means, α and β are computed.2
Now suppose that a new computational domain of size NL, where
N is a largish positive integer, is constructed by replication of the
original domain. Clearly, in the absence of perturbation, the solution
of the extended system will continue to evolve as N 3 replicas of
the original system. If the extended system is subject to a long
wavelength perturbation then it will relax to a new state, which, in
general, is not periodic on scales smaller than NL; in other words, the
system will transfer some of its energy to scales with wavenumbers
smaller than 2π/L. The question is, will the coefficients α and β
capture any aspect of this relaxation process? In particular, will the
growth or decay of modes with (small) wavenumber k be captured
by expression (1)? The answer, in general, is no. The final state could
be similar to the initial state (i.e. that obtained from replicating the
smaller domain) or could be very different with, say, considerable
energy at large scales. However, irrespective of the nature of the
final state, the relaxation process by which it is achieved depends
on the non-linear interactions between all the scales larger than L,
and is therefore not captured by averages over the scale L. Once
again, the non-linear transfer of excitations to intermediate scales
invalidates the necessary assumption of scale separation. Thus, in
this case also, although α and β can be measured, they do not convey
useful information about magnetic field evolution.
AC K N OW L E D G M E N T S
We are grateful to Karl-Heinz Rädler and the referee for their helpful
comments, which led to the improvement of the Letter. This research
was supported by the National Science Foundation sponsored Center for Magnetic Self Organization (CMSO) at the University of
Chicago, and by the Science and Technology Facilities Council.
The Letter was completed at the Kavli Institute for Theoretical
Physics, supported by Grant No. NSF PHY05-51164.
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The problem is that we are interested in calculating α and β as properties of
the saturated non-linear state. The measurement technique should not perturb
the flow to a different state. Therefore, of the techniques we described in
Section 3, probably only the Lagrangian approach can be guaranteed to
achieve this result.
This paper has been typeset from a TEX/LATEX file prepared by the author.